Seems you also lose one level of Heat-Up if your Drive move is blocked. http://youtu.be/ozccIvsKmaQ?t=5m48s (In this vid. she's in H2, jump near Plat. and try to hit her with a j.D, but is blocked and return to H1.)
This means you better use the Drive in a combo to avoid losing Heat-Up. It's regrettable since H2 Drive gives you a fast way to close-in. I guess if you're in H0 you can still use Drive for pressure at close range.
About Overdrive, it's rare to see players using it at the start of the round just to gain safely the H1. I remember one guy trying the tech during the first days, then nothing. Do you think it's not worth spending the Overdrive/Burst meter for this ? Normally you regain the icon after ~40 seconds, thus you could Overdrive+Burst in the same round. I saw some Hakumen players using this technique to gain 2 or 3 extra Magatama at the start of the round.